Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{3} + 5 x + 57 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 2\cdot 59 + 20\cdot 59^{2} + 59^{3} + 28\cdot 59^{4} + 41\cdot 59^{5} + 56\cdot 59^{6} + 4\cdot 59^{7} + 50\cdot 59^{8} + 16\cdot 59^{9} + 19\cdot 59^{10} + 44\cdot 59^{11} + 55\cdot 59^{12} +O\left(59^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 a^{2} + 43 a + 28 + \left(19 a^{2} + 23 a + 45\right)\cdot 59 + \left(5 a^{2} + 17\right)\cdot 59^{2} + \left(43 a^{2} + 10 a + 45\right)\cdot 59^{3} + \left(21 a^{2} + 3 a + 52\right)\cdot 59^{4} + \left(18 a^{2} + 45 a + 21\right)\cdot 59^{5} + \left(30 a^{2} + 38 a + 22\right)\cdot 59^{6} + \left(50 a^{2} + 7 a + 50\right)\cdot 59^{7} + \left(15 a^{2} + 28 a + 52\right)\cdot 59^{8} + \left(10 a^{2} + 26 a + 53\right)\cdot 59^{9} + \left(20 a^{2} + 36 a + 27\right)\cdot 59^{10} + \left(10 a^{2} + 57 a + 34\right)\cdot 59^{11} + \left(44 a^{2} + 44 a + 9\right)\cdot 59^{12} +O\left(59^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 a^{2} + 25 a + 44 + \left(51 a^{2} + 52 a + 33\right)\cdot 59 + \left(53 a^{2} + a + 2\right)\cdot 59^{2} + \left(22 a^{2} + 22 a + 37\right)\cdot 59^{3} + \left(17 a^{2} + 18\right)\cdot 59^{4} + \left(23 a^{2} + 29 a + 38\right)\cdot 59^{5} + \left(21 a^{2} + 52 a + 51\right)\cdot 59^{6} + \left(40 a^{2} + 51 a + 55\right)\cdot 59^{7} + \left(7 a^{2} + 54 a + 5\right)\cdot 59^{8} + \left(15 a^{2} + 20 a + 11\right)\cdot 59^{9} + \left(41 a^{2} + 56 a + 39\right)\cdot 59^{10} + \left(10 a^{2} + 13 a + 35\right)\cdot 59^{11} + 12 a\cdot 59^{12} +O\left(59^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a^{2} + 50 a + 46 + \left(47 a^{2} + 41 a + 38\right)\cdot 59 + \left(58 a^{2} + 56 a + 38\right)\cdot 59^{2} + \left(51 a^{2} + 26 a + 35\right)\cdot 59^{3} + \left(19 a^{2} + 55 a + 46\right)\cdot 59^{4} + \left(17 a^{2} + 43 a + 57\right)\cdot 59^{5} + \left(7 a^{2} + 26 a + 43\right)\cdot 59^{6} + \left(27 a^{2} + 58 a + 11\right)\cdot 59^{7} + \left(35 a^{2} + 34 a\right)\cdot 59^{8} + \left(33 a^{2} + 11 a + 53\right)\cdot 59^{9} + \left(56 a^{2} + 25 a + 50\right)\cdot 59^{10} + \left(37 a^{2} + 46 a + 47\right)\cdot 59^{11} + \left(14 a^{2} + a + 48\right)\cdot 59^{12} +O\left(59^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 57 a^{2} + 45 a + 13 + \left(56 a^{2} + 53 a + 52\right)\cdot 59 + \left(19 a^{2} + 38 a + 46\right)\cdot 59^{2} + \left(43 a^{2} + 6 a + 6\right)\cdot 59^{3} + \left(28 a^{2} + 50 a + 17\right)\cdot 59^{4} + \left(6 a^{2} + 37 a + 41\right)\cdot 59^{5} + \left(29 a^{2} + 51 a + 57\right)\cdot 59^{6} + \left(4 a^{2} + 25 a + 14\right)\cdot 59^{7} + \left(15 a^{2} + 31 a + 50\right)\cdot 59^{8} + \left(23 a^{2} + 34 a + 57\right)\cdot 59^{9} + \left(36 a^{2} + 26 a + 22\right)\cdot 59^{10} + \left(28 a^{2} + 10 a + 36\right)\cdot 59^{11} + \left(12 a^{2} + 25 a + 41\right)\cdot 59^{12} +O\left(59^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a^{2} + 13 a + 18 + \left(25 a^{2} + 9 a + 6\right)\cdot 59 + \left(2 a^{2} + 19 a + 8\right)\cdot 59^{2} + \left(5 a^{2} + 54 a + 56\right)\cdot 59^{3} + \left(28 a^{2} + 30 a + 14\right)\cdot 59^{4} + \left(19 a^{2} + 36 a + 45\right)\cdot 59^{5} + \left(27 a^{2} + 45 a + 51\right)\cdot 59^{6} + \left(22 a^{2} + 23 a + 15\right)\cdot 59^{7} + \left(22 a^{2} + 47 a + 35\right)\cdot 59^{8} + \left(26 a^{2} + 31 a + 48\right)\cdot 59^{9} + \left(11 a^{2} + 3 a + 57\right)\cdot 59^{10} + \left(39 a^{2} + 23 a + 51\right)\cdot 59^{11} + \left(40 a^{2} + 29 a + 56\right)\cdot 59^{12} +O\left(59^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 31 + 53\cdot 59 + 12\cdot 59^{2} + 37\cdot 59^{3} + 51\cdot 59^{4} + 8\cdot 59^{5} + 47\cdot 59^{6} + 58\cdot 59^{7} + 8\cdot 59^{8} + 50\cdot 59^{9} + 44\cdot 59^{10} + 35\cdot 59^{11} + 15\cdot 59^{12} +O\left(59^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 32 a^{2} + a + 28 + \left(35 a^{2} + 55 a\right)\cdot 59 + \left(36 a^{2} + 4\right)\cdot 59^{2} + \left(10 a^{2} + 57 a + 55\right)\cdot 59^{3} + \left(2 a^{2} + 36 a + 26\right)\cdot 59^{4} + \left(33 a^{2} + 43 a + 31\right)\cdot 59^{5} + \left(2 a^{2} + 20 a + 8\right)\cdot 59^{6} + \left(32 a^{2} + 9 a + 28\right)\cdot 59^{7} + \left(21 a^{2} + 39 a + 32\right)\cdot 59^{8} + \left(9 a^{2} + 51 a + 11\right)\cdot 59^{9} + \left(11 a^{2} + 28 a + 37\right)\cdot 59^{10} + \left(50 a^{2} + 25 a + 29\right)\cdot 59^{11} + \left(5 a^{2} + 4 a + 19\right)\cdot 59^{12} +O\left(59^{ 13 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 24 + 3\cdot 59 + 26\cdot 59^{2} + 20\cdot 59^{3} + 38\cdot 59^{4} + 8\cdot 59^{5} + 14\cdot 59^{6} + 54\cdot 59^{7} + 58\cdot 59^{8} + 50\cdot 59^{9} + 53\cdot 59^{10} + 37\cdot 59^{11} + 46\cdot 59^{12} +O\left(59^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,9,7)(2,4,3)(5,8,6)$ |
| $(1,9)(2,3)(5,6)$ |
| $(1,6)(5,9)(7,8)$ |
| $(1,7,9)(5,8,6)$ |
| $(1,3,6)(2,5,9)(4,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,9)(2,3)(5,6)$ |
$0$ |
| $9$ |
$2$ |
$(1,6)(5,9)(7,8)$ |
$-2$ |
| $9$ |
$2$ |
$(1,5)(2,3)(6,9)(7,8)$ |
$0$ |
| $2$ |
$3$ |
$(1,9,7)(2,4,3)(5,8,6)$ |
$-3$ |
| $6$ |
$3$ |
$(1,3,6)(2,5,9)(4,8,7)$ |
$0$ |
| $6$ |
$3$ |
$(1,7,9)(2,4,3)$ |
$0$ |
| $12$ |
$3$ |
$(1,5,2)(3,7,6)(4,9,8)$ |
$0$ |
| $18$ |
$6$ |
$(1,2,6,9,3,5)(4,8,7)$ |
$0$ |
| $18$ |
$6$ |
$(1,5,7,6,9,8)(2,4,3)$ |
$1$ |
| $18$ |
$6$ |
$(1,2,7,4,9,3)(6,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
